The cross-domain representation learning plays an important role in tasks including domain adaptation and transfer learning. However, existing cross-domain representation learning focuses on building one shared space and ignores the unlabeled data in the source domain, which cannot effectively capture the distribution and structure heterogeneities in cross-domain data. To address this challenge, we propose a new cross-domain representation learning approach: MUltiple Lipschitz-constrained AligNments (MULAN) on partially-labeled cross-domain data. MULAN produces two representation spaces: a common representation space to incorporate knowledge from the source domain and a complementary representation space to complement the common representation with target local topological information by Lipschitz-constrained representation transformation. MULAN utilizes both unlabeled and labeled data in the source and target domains to address distribution heterogeneity by Lipschitz-constrained adversarial distribution alignment and structure heterogeneity by cluster assumption-based class alignment while keeping the target local topological information in complementary representation by self alignment. Moreover, MULAN is effectively equipped with a customized learning process and an iterative parameter updating process. MULAN shows its superior performance on partially-labeled semi-supervised domain adaptation and few-shot domain adaptation and outperforms the state-of-the-art visual domain adaptation models by up to 12.1%.